Category Archives: normativity

Thinking about an experiment on “practical math” in normative contexts

I am trying to think about an experimental situation that would allow me to test how normative-institutional contexts impact on children quantitative reasoning. Ideally, it has to be an easy experimental task that can be tested quickly with children from different cultures. What follows is a half-baked draft. Your feedback and criticism is most welcome.

So this is the idea… Children (ages 4 to 7) are interviewed individually. During the interview, they are shown a series of very short puppet plays. After each play children are questioned about the best way to solve a problem that arose in the play. Children are required to offer quantitative answers to such problems; for example, “how much money does character A have to pay character B to get even?” or “How many blocks does character A need to add in order to complete the building?”, etc.  The narratives are different in nature. Some narratives provide a social and normative context to the problem, in the sense that they highlight certain social rules children need to take into account in order to respond appropriately to the situation. Other narratives, by way of contrast, highlight “technical” or “engineering” problems, and involve means-ends reasoning. They problems they involve are similar to the normative problems in their mathematical content, yet the narrative context is markedly different.


A1: “Negative reciprocity and reparation”. Character A has a bag with three candy bars. Character A shows the bag to character B and tells her that she loves candy bars and that she plans to eat them with her friends the next day. Character A goes to sleep. Character B steals the bars and eats them. Character A wakes up and finds character B stole the candy, and asks character B to return them. Character B says she doesn’t have the candy anymore but that she can offer character A some money to make up for the stolen candy. She opens a purse and drops some coins and bills on the table. The child is asked to choose the coins and bills character B has to hand over to character A in order to get even. They child is questioned about how she made that decision; and how she calculated how many bills and coins that character B must give character A.

A2: “Destruction and reconstruction”. The child is shown a tower formed by six big blocks. The child is told that a powerful storm and strong winds hit the building during the night and broke the three upper stories of the building. She’s then given a number of smaller blocks of different sizes and is asked to rebuild the tower so that it is as high as it was before the storm. The child is questioned the criteria she used to select the blocks and to decide how many blocks to use.

B1. “Positive reciprocity”. Character A visits character B and shows up with a present: a stack of stickers or trading cards. Each character returns to her own home. Then character B says that character A was really nice and that she would also like to give her a present to “get even”. The child is asked to help character B prepare her present. She is shown a cup and a collection of marbles and is told to fill the cup up until there are enough for A’s present. The child is also asked about how she decided how many marbles to give; i.e. to justify her decision.

B2. “Bridging the gap”. The child is shown a model of a river. On the river there is half bridge built with legos. The bridge starts on one shore and goes only half-way over the river. The child is asked to pick the lego pieces that she would need to build the other half of the bridge. The set of lego pieces the child can choose from have a different size than the ones used to build the first half of the bridge.

All four situations involve some kind of addition and subtraction of different units; they also involve compensating different dimensions of problems (values of the goods exchanged, sizes of different objects, etc.) A1 and B1 are “social” and “normative”: they involve the concept of justice; A2 and B2 are “technical”: they involve a kind of means-ends reasoning.

One possibility is to give situations A1-B1 to one group and A2-B2 to a different group. One could then compare the reasoning and argumentation of children who are given a “normative” vis- à-vis a “technical” narrative. To this end, one might use the theory of argumentation and other tools of discourse analysis. One could also do some standardized numeracy tests (perhaps those used by Opfer & Siegler, Dehaene, Piagetian conservation tests, etc.) after the main tasks in order to evaluate if each of these normative contexts has “sensitized” the child to quantities in a special way; i.e. if the children who just completed the “technical” problem perform better or worse than the children who did the “social-normative” problem.

Another possibility is to give the same children all four situations so as to compare the features of quantitative thinking in technical vs. normative contexts in the same children.

Still thinking…

The normativity of human knowledge

I am now reading Prof. Castorina’s lectures on Genetic Epistemology. There he makes the case that human knowledge in general, and scientific knowledge in particular, involves a normative dimension that is often overlooked by naturalistic approaches to knowledge.

Let me explain this topic in my own words. Naturalized Epistemology is right in considering human knowledge as a fact of the world. Human beings are real, corporeal, natural entities. Human beings have (are) bodies; they have a physical existence. Any explanation of human knowledge must recognize that humans can know their world only insofar as they are equipped with wet computers (aka brains) that receive information from the world, process it, and respond to the world in a certain manner. There’s input, information processing and output. If your computer gets broken (in a serious car accident, for example), you might lose your ability to know the world.

Although I am already using a highly metaphorical language here (because the brain is different from a digital computer in many significant ways), I can buy the previous description up to this point. Human knowledge is a natural phenomenon and therefore it can be studied by using the methods of the natural sciences (for example, the neurosciences).

Yet when we look at actual human beings engaged in knowledge-related practices (human beings investigating, thinking, theorizing, teaching, learning and discussing about different issues) an important aspect of human knowledge comes to light. Not only do people know about certain things, they also know that what they know is true. For instance, they know that the sentence “dogs are mammals” is true; and they can defend the truth of such a claim through arguments. People can (and frequently do) justify most of their knowledge claims. They offer reasons why things are in a certain (and not in another) way. They argue for specific positions. They follow rules and shared criteria for adjudicating between rival hypotheses. They claim that some assertions are true and they also claim to know why they are true. In certain cases (two plus two equals four) most human beings would argue that the truth of this claim is universal and necessary. That is, they would say that they know not only that things are in a certain way, but also why they must be that way and couldn’t possibly be in any other way.

To put it differently: people care not only about the efficacy of their knowledge (whether what they know allows them to adapt effectively to the external reality) but also about the legitimacy of their knowledge. Any observation of actual human beings involved in knowledge-related practices makes this point self-evident. Any observation of naturalistic epistemologists giving talks in conferences or workshops or making arguments to convince others makes this point self-evident. They are not just blind mechanisms sputtering output; they try to be rational, sensible, persuasive.

There is a normative dimension to human knowledge. The problem with the naturalistic approach to human knowledge is that it cannot bridge the gap between the mechanistic – naturalistic level of explanation and the normative phenomena. What humans know is not just the result of some material mechanism (involving the interaction between the world and the wet computer) but is also the result of a complex socio-cultural normative process that requires to be addressed on a different level. The natural sciences by themselves cannot account for this normative component; norms and institutions must be included.

Epistemology, therefore (and this is Castorina’s point) should deal with the fundamental problem of how people and societies give themselves norms. Any relevant epistemology must start by recognizing the normativity of human knowledge.